Irrationality of Catalan's constant

Theorem 1.1: $G$ is irrational. Under the hypothesis $G\in\mathbb Q$ the paper builds mixed determinants $\Delta_N$ of size $48N$ from moments of two kernels, whose $\zeta(2)$ terms cancel by Taylor contact of Chebyshev rows, so all $\Delta_N$ are rational; a finite certificate gives $\Delta_p\ne0$ for large primes $p$, prime-by-prime denominator bounds give $\liminf\mathcal L_p>-2.29084$, and a real-place integral estimate gives $\limsup\mathcal L_N<-2.2909$, a contradiction. Corollaries: $4G$, the minimal volume of a two-cusped orientable hyperbolic 3-manifold, and every arithmetic hyperbolic 3-orbifold volume over $\mathbb Q(i)$ are irrational. Not shown: an irrationality measure, transcendence, or anything about $\beta(4),\beta(6),\ldots$. An earlier preprint by Zhi-Wei Sun (arXiv:2609.04176, 3 September 2026) also claims irrationality; the manuscript says it uses nothing from it, so the first claimed proof may not be this one.

Theorem 1.1: $G$ is irrational. Under the hypothesis $G\in\mathbb Q$ the paper builds mixed determinants $\Delta_N$ of size $48N$ from moments of two kernels, whose $\zeta(2)$ terms cancel by Taylor contact of Chebyshev rows, so all $\Delta_N$ are rational; a finite certificate gives $\Delta_p\ne0$ for large primes $p$, prime-by-prime denominator bounds give $\liminf\mathcal L_p>-2.29084$, and a real-place integral estimate gives $\limsup\mathcal L_N<-2.2909$, a contradiction. Corollaries: $4G$, the minimal volume of a two-cusped orientable hyperbolic 3-manifold, and every arithmetic hyperbolic 3-orbifold volume over $\mathbb Q(i)$ are irrational. Not shown: an irrationality measure, transcendence, or anything about $\beta(4),\beta(6),\ldots$. An earlier preprint by Zhi-Wei Sun (arXiv:2609.04176, 3 September 2026) also claims irrationality; the manuscript says it uses nothing from it, so the first claimed proof may not be this one.